Given a lattice \(\Lambda \subset \mathbb {R}^n\)  of rank \(n\) , Algorithm REF  enables
to find the Boolean equation of a coordinate \(\hat{z}_i\) .
It also finds the equation of each hyperplane needed to get the value of the Boolean variables involved in the equation.
This algorithm can be seen as a “training” step to “learn” the structure of the lattice.
It is a brute-force search that may quickly become too complex as the dimension increases. However, we shall see in Section REF  and REF  that these Boolean equations can be analyzed without this algorithm, via a study of the basis.
Note that the decoding complexity does not depend on the complexity of this search algorithm.

Brute-force search to find the Boolean equation of a coordinate \(\hat{z}_i\)  for a lattice \(\Lambda \)

[1]
Select the \(2^{n-1}\)  corners of \(\mathcal {P}(\mathcal {B})\)  where \(z_{i}=1\)  and all relevant Voronoi vectors of \(\Lambda \) .
each of the \(2^{n-1}\)  corners where \(z_{i}=1\) 
each relevant Voronoi vector of \(\Lambda \) 

Move in the direction of the selected relevant Voronoi vector by half its norm + \(\epsilon \)  (\(\epsilon \)  being a small number).

The resulting point is outside \(\mathcal {P}(\mathcal {B})\) .
Do nothing. //There is no decision boundary hyperplane in this direction.
Find the closest lattice point \(x^{\prime }=z^{\prime }G\)  (e.g. by sphere decoding [1]}).
\(z^{\prime }_{i}=1\) 
Do nothing. //There is no decision boundary hyperplane in this direction.
Store the decision boundary orthogonal to this direction. //\(z^{\prime }_{i}=0\) 
each decision boundary hyperplane found (at this corner)
Associate and store a Boolean variable to this hyperplane (corresponding to the position of the point to be decoded with respect to the hyperplane).
The Boolean equation of \(\hat{z}_i\)  contains a Boolean AND of these variables.
The equation is the Boolean OR of the \(2^{n-1}\)  AND coming from all corners.
